Although I haven't done much long riding, I know a few people who do and I do spend as much time as possible in the saddle, so I have some thoughts here.
The answers that say that a horse travels on average at brisk walking speed for a human are basically correct- you can actually make somewhat better time if you trot some of the time, cantering/galloping everywhere in the way that people do in films or video games is entirely inaccurate. On average it is a little faster than travelling on foot, but the biggest advantage is that the rider is not as tired at the end of it.
Bear in mind that horses are very capable of carrying a rider a long distance, but if you are wanting to also carry supplies, camping equipment and any other needs for the road then you will probably want to have some pack animals along. These may be horses or ponies but there would be a good chance of donkeys or mules being used for pack too.
The biggest problems that you will run into on long journeys are likely to be to do with the comfort of the equipment over time- a well fitted saddle will be comfortable enough over time, but any problems with saddle fit quickly tell on the horse's ability and willingness to keep moving. Likewise there are many reasons that lameness can arise from muscle strains or hoof abscesses to much more serious injuries. A considerate rider will get off the moment a horse shows any sign of lameness and if it seems serious that would probably be the end of the day's travel. Horses are surprisingly fragile.
The quality of feed is also important- a sudden change in richness risks a horse colicking, which is often fatal. Likewise mouldy or bad feed is liable to result in colic. Too much energy ( typically too much grain ) and you'll find yourself sat on an unexploded bomb - although once you're engaged on a long journey like this, then higher-energy feed in smaller portions will help to maintain strength.
In terms of the journey there is also a consideration of how you keep the horses from wandering off overnight- tethering them individually may not be practical so I would expect to either picket or hobble them. Hobbling ties up a leg ( or ties too legs together ) so that horses can move but not very fast and they're unlikely to go far away. If you have a well established herd, you would probably only need to hobble the lead mare and the others would be unlikely to stray far from her. However it is not unknown to have to spend a while tracking down hobbled horses in the morning if something has spooked them or it smelled like some interesting food off in the distance. Picketing - tying the horses to a line between two trees or posts - keeps them exactly where you know they are, but they can't browse during the night beyond the feed they are given. Horses are trickle-feeders by nature so it is healthiest for them to be grazing as much as possible- allowing them to browse overnight fulfills this need. Some useful information on setting up camp with horses.
You might imagine that carrying a rider from a horse's point of view is a little like our experience of carrying a heavy backpack. There are lots of ways that even a well fitted saddle can cause problems over time ( especially if the rider is not expertly balanced ) and an experienced rider will make sure they lift up the saddle from time to time to allow the back to cool down and get some ventilation but also on a long journey you would probably want to give horses at least one day off every week or so, just to give them a chance to stretch and recover from their travels.
It may be worth reading some accounts of long distance horse travel- I recommend The Fairly Big Ride as being entirely online and a really enjoyable read.